Question about ideals of a ring: $IJ=I \implies J=I$?

Question

Doing exercises, this question came to my mind.

1. Is it true that if $I$ and $J$ are proper and nonzero ideals of a ring $R$, $$IJ=I \implies I=J?$$

2. And $$IJ=I \iff I\subseteq J?$$

Answer 1

Automatically if $I=IJ$ then $I=IJ\subseteq J$ i.e. $I\subseteq J$. But equality isn't necessarily true.

Consider $R=F\times F\times F$ and the ideals $I$ and $J$ generated by $(1,0,0)$ and $(1,1,0)$.

Moreover, equality can occur when $I=I^2$ but can't occur if $I\ne I^2$.